The use of a rare sense of “exact” confuses people, but the traditional phrase is “exact revenge”, not the seemingly more logical “extract revenge” or “enact revenge.”
Hamilton softball falls in state semifinals QUEENSBURY >> The last time Hamilton and Afton met on the softball field coach Bill Dowsland’s Emerald Knights posted a 1-0 upset win in a state regional game. Saturday afternoon, nine long years later, Afton was able to exact revenge with a 1-0 victory in the Class D state semifinals.
June 9, 2013 - The Oneida Daily Dispatch
'Walking Dead' actress arrested: How ricin letters became tool of revenge Several poisonous letters sent to officials appear to have been tied to settling private feuds. On Friday, the FBI arrested a woman for sending ricin-tainted letters to President Obama and New York Mayor Michael Bloomberg, apparently to exact revenge upon her estranged husband.
June 8, 2013 - The Christian Science Monitor